Getting a SSD on your iMac, even if external, would make it feel like a totally different machine. I highly recommend.
Also, you might get some replies saying not to bother adding a SSD or doing anything to such an old device, but I think it is silly to spend $2000+ on a new Mac because your HDD of you current Mac is slow, when a small investment $50 (500GB) to $100 (1TB) could get you a few more years out of your machine.
The cheapest option would be a SATA SSD and a USB3 enclosure or USB/SATA adapter cable. While it would be faster to do an internal SSD for HDD swap, the USB SSD would still be a night and day difference from your aging HDD.
You can currently get a 1TB SATA SSD on New Egg for $63 and a 500GB SATA SSD for $40. Those are the cheapest options if money is a factor, so if you want something more name brand, the prices will go up some.
Best Buy has PNY on sale right now, $46 for 500G and $78 for 1TB.
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USB/SATA adapter cables are around $10, I like and own a few of these because the cable is a little longer:
Product Description:Take advantage of the lightening speed of USB 3.0 to quickly transfer files up to 5 Gbps! Connect any 2.5'' Serial ATA (SATA) / SSD or SATA II Hard Drive to your computer through an available USB 3.0 port on your desktop or laptop. Reverse compatible with USB 2.0 and 1.1, this...
www.amazon.com
If you feel more comfortable with an enclosure, they are also around $10.
It doesn't take much skill to set up, and can be done pretty quickly. Just install the OS of choice, and use Apple's Migration Assistant built into MacOS to migrate your apps, files, photos, etc. over to your new drive.
Side Note: It is good to have a BU of your current drive, just general good practice. Also, I would update the internal drive to High Sierra, regardless of what OS you plan on using, this will update the firmware.
Would you consider an external SSD on thunderbolt or on Usb3?
There is a bunch of options for you:
If money is an issue, then a USB3 drive is the way to go. You get great speed compared to you old HDD, the "instant" feel of an SSD, and it can happen for the price of a weeks worth of Starbucks.
If you plan on using this SSD for many years, TB/TB2 might be a better choice do to TRIM support which is not available for USB drives on MacOS. Finding TB/TB2 drives is harder than TB3 and much harder than USB. The cheapest way for this is to get a used TB HDD on eBay and swap the SATA HDD for a SATA SSD. Not difficult, but usually the cheapest way to get a TB SSD.
If you got the need for speed, a much more expensive option is TB3 NVMe. Depending on whether the TB3 drive has its own power source or not, this could involve investment in additional equipment to power the drive, such as a TB3 dock.
There is even more complex options, such as striping RAIDs, but I won't get into that unless you are interest.
Why are you considering high sierra rather than Catalina?
I wouldn't consider Catalina if you are using older, unsupported SW. So it depends on what you are doing with your Mac.
High Sierra runs great on the Late 2013 iMacs, and is a pretty good choice. It looks modern, while still be compatible with older SW. It no longer gets security updates, but using a third party web browser would cover a lot of that security. Firefox tends to support MacOS long after Apple has given up on it.
Catalina still gets security updates, so if that is important to you, that would be the way to go. If you plan on using your HDD as a bootable BU drive for your external SSD, I wouldn't use Catalina. APFS and HDDs do not work well together. If you are using your old HDD as extra storage or not using it at all, then Catalina would be fine.